In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu et. al. [13]. We focus primarily on products form a semi-direct product. First, twisted representatives right-angled Artin and Coxeter groups. prove regularity geodesic language for certain cases, highlight properties spherical language, depending automorphism ordering generating ...

Consider a smooth connected solvable group G over a field k. If k is algebraically closed then G = T nRu(G) for any maximal torus T of G. Over more general k, an analogous such semi-direct product structure can fail to exist. For example, consider an imperfect field k of characteristic p > 0 and a ∈ k−kp, so k′ := k(a1/p) is a degree-p purely inseparable extension of k. Note that k′ s := k ′ ⊗k...